SOLUTION: use the given table of values to estimate the volume of the solid formed by revolving y = f(x), {{{0 <= x <= 3}}}, about the x-axis, the table :- ___________ __________

Plot the 7 points and connect them:



When we rotate it about the x axis, we'll have a solid figure
with this mid-cross section:



It says just to estimate the volume.  So it may be that your teacher
just wants you to average up those 7 y-values like this:

 = 

and then assume the volume is about the same volume as a cylinder with 
a radius of  like the cylinder which the green lines 
represent a mid-cross section of below:




If that's the kind of estimate your teacher wants, then we
just use the volume of a cylinder with that average y-value
as a its radius.  The cylinder's height h (measured horizontally)
is 3 units,

The formula for the volume of a cylinder is

V =  =  = 35.57565167

So that ought to be a pretty good estimate, about 36 cubic units.

Maybe that's all your teacher wants.

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Just for fun, let's find the true volume: and compare:

We'll divide that mid-cross section into 6 little trapezoids:



Each of those trapezoids is a mid cross section of a frustrum of
a cone, with height h, and the right and left radii are the
values of f(x).  The formula for the volume of a frustrum of a
cone is:



The height of each one is  so the above will become:





Let the volumes of the frustrums of the cones be V1,...,V6

Total volume = 





 







Since all are multiplied by 

Total volume = 

 = 

That rounds to 36 cubic units, so the estimate using the cylinder was pretty good.

Edwin